BIREP – Representations of finite dimensional algebras at Bielefeld

• BIREP
• Faculty of Mathematics
• Bielefeld University

 

Seminar

No talks have been announced for his week.

For a regular email announcement please contact birep.

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Future Talks

Friday, 08 June 2012

• Lecture Hall H10
  Shawn Baland (Aberdeen): The generic kernel filtration for modules of constant Jordan type
  Abstract: Let E be an elementary abelian p-group of rank two and k an algebraically closed field of characteristic p. Recently, Carlson, Friedlander and Suslin have constructed a canonically defined submodule of a kE-module called the generic kernel. In the case where M is a kE-module of constant Jordan type, they have shown that the generic kernel admits a filtration of M in which many of the terms have constant Jordan type. In this talk I will introduce a duality formula for subquotients in the above filtration and answer the authors' question regarding whether or not all filtration terms have constant Jordan type.
• Lecture Hall H10
  Johan Steen (Trondheim): tba

Friday, 15 June 2012

• Lecture Hall H10
  Raquel Simoes (Leeds): tba
• Lecture Hall H10
  Yong Jiang (Bielefeld): Every projective variety is a quiver Grassmannian (after Reineke)

Friday, 22 June 2012

• Lecture Hall H10
  Martin Kalck (Bonn): (Relative) Singularity categories
  Abstract: There are two interesting triangulated categories associated with any (MCM-representation finite) Gorenstein singularity: the singularity category of Buchweitz and the relative singularity category of a non-commutative (Auslander) resolution, which was studied in joint work with Burban. We show that these categories mutually determine each other in the case of ADE-singularities in any Krull dimension. Knörrer's periodicity theorem yields a wealth of non-trivial examples. This is joint work with Dong Yang.
• Lecture Hall H10
  Reiner Hermann (Bielefeld): tba

Friday, 29 June 2012

• Lecture Hall H10
  Dong Yang (Stuttgart): tba
• Lecture Hall H10
  Daniel Labardini-Fragoso (Bonn): tba
• 16:00, Lecture Hall H10
  Christian Stump (Hannover): Revisiting the combinatorics of cluster categories in finite types
  Abstract: I will present the natural combinatorial construction of subword complexes and describe their connections to repetition quivers, Auslander-Reiten quivers, and cluster categories in finite types. This perspective leads to new combinatorial objects called multi-cluster complexes, of which I will discuss possible relations to certain identifications in the repetition quiver in terms of Auslander-Reiten translates and shifts.

Friday, 06 July 2012

• Lecture Hall H10
  Antoine Touzé (Paris): Finite generation of the cohomology of reductive group schemes
  Abstract: It has been proved by Haboush that if G is a reductive group scheme acting on an algebra A commutative and finitely generated, then the invariants A^G form a finitely generated algebra (this was conjectured by Mumford).
  Evens (1964), and later Friedlander Suslin (1997), proved similar finite generation theorems for the cohomology ring H^*(G,A) for finite groups and finite group schemes G acting on a commutative finitely generated algebra A. After this, van der Kallen conjectured that all the reductive group schemes have finitely generated cohomology algebras.
  In this talk we will present an overview of the proof of this theorem and how strict polynomial functors come into play for this problem.
  NB: The finite generation result depends (directly or indirectly) on the contribution of many authors: Friedlander, Suslin, van der Kallen, Srinivas, Grosshans and Touzé, and a complete proof is available in the article:
  "Bifunctor cohomology and Cohomological finite generation for reductive groups" Duke Math. J.  151  (2010).
• Lecture Hall H10
  Ulrich Krähmer (Glasgow): Batalin-Vilkovisky Structures on Ext and Tor
  Abstract: The topic of this talk (based on joint work with Niels Kowalzig) is an algebraic structure whose best known example is provided by the multivector fields and the differential forms on a smooth manifold: the multivector fields are a Gerstenhaber algebra with respect to wedge product and Schouten-Nijenhuis bracket, and the differential forms are what we call a Batalin-Vilkovisky module over this Gerstenhaber algebra, which means that the multivector fields act in two ways on forms - by means of contraction and of Lie derivative - and that these actions are related by a differential that fits into Cartan's "magic" homotopy formula. Nest, Tamarkin and Tsygan suggested to refer to this abstract package of two graded vector spaces with such operations as to a noncommutative differential calculus.
  Generalising work by the above mentioned authors and by Getzler, Gerstenhaber, Goodwillie, Huebschmann, Rinehart and others, I will explain that Ext and Tor over Hopf algebroids tends to carry such a structure which means that homological algebra produces plenty of examples of noncommutative differential calculi, including for example Hochschild and Poisson (co)homology.
  As a first application, Ginzburg's theorem that the Hochschild cohomology ring of a Calabi-Yau algebra is a Batalin-Vilkovisky algebra is extended to twisted Calabi-Yau algebras such as quanum groups, quantum homogeneous spaces or quantum vector spaces.
• Lecture Hall H10
  Werner Hoffmann (Bielefeld): tba

Friday, 13 July 2012

• Markus Szymik (Bochum/Düsseldorf): tba

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Seminar Archive

Grzegorz Bobinski made his notes from some of the seminar talks available on his web page.

Friday, 04 May 2012

• 14:15, Lecture Hall H10
  Nils Mahrt (Bielefeld): Idempotents in Representation Rings of Quivers
  Abstract: This is a report on work of Herschend, Kinser and Schiffler. Let Q be a quiver and k an algebraically closed field. We will define the representation ring R(Q) as follows: On the split Grothendieck group of the category of representations of Q define a multiplication induced by the pointwise tensor product of quiver representations. For an acyclic quiver Q we will construct certain orthogonal idempotent elements in R(Q).

Friday, 20 April 2012

• 14:15, Lecture Hall H10
  David Pauksztello (Hannover): Co-t-structures and co-stability
  Abstract: In this talk we introduce the ideas of co-t-structures and co stability conditions and compare and contrast with t-structures and stability conditions. We show that the space of co-stability conditions on a triangulated category forms a complex manifold, and give some examples.

Friday, 13 April 2012

• 14:15, Lecture Hall H10
  Claus Michael Ringel (Bielefeld): Morphisms determined by objects: The case of modules over artin algebras
  Abstract: Let R be an artin algebra. In his Philadelphia Notes, Auslander showed that any homomorphism between R-modules is right determined by an R-module C, but a formula for C which he wrote down has to be modified. The lecture will discuss the indecomposable direct summands of the minimal right determiner of a morphism, in paricular the role of the projective direct summands. We will provide a detailed analysis of those morphisms which are right determined by a module without any non-zero projective direct summand. What we encounter is an intimate relationship to the vanishing of Ext^2.

Friday, 23 March 2012

• 13:15, Room U2-205
  Jan Schröer (Bonn): Generic Caldero-Chapoton functions and generalized clusters
  Abstract: This is joint work with Giovanni Cerulli Irelli and Daniel Labardini-Fragoso. To any algebra A defined as a factor of a completed path algebra we associate an algebra CC(A) generated by generalized Caldero-Chapoton formulae. We construct a candidate G(A) for a basis of CC(A), and we show that G(A) is linearly independent. All elements in G(A) are generalized cluster monomials. Then we apply these results to the theory of cluster algebras. Even for acyclic cluster algebras one obtains a much richer structure than the classical cluster algebra theory can provide. We use the concept of strongly reduced components of module varieties introduced by Geiss, Leclerc and Schröer, and generalizations of recent results by Plamondon, who parametrizes strongly reduced components for finite-dimensional algebras.
• 14:30, Room U2-205
  Dave Benson (Aberdeen): Around a theorem of Mislin in the cohomology of finite groups
  Abstract: Mislin proved in 1990 that the inclusion of a subgroup H in a finite group G induces an isomorphism in mod p cohomology if and only if the index is prime to p and H controls fusion in G. His proof was essentially topological in nature, as it used the then recent proof of the Sullivan conjecture as well as results of Dwyer and Zabrodsky. Peter Symonds partly algebraised the proof in 2004, and the rest of the proof was recently algebraised by Hida and Okuyama in terms of some rather intricate arguments with cohomology of trivial source modules.
  It turns out that if p is odd, a much stronger statement is true. Namely, if |H:G| is prime to p and the inclusion just induces an F-isomorphism in mod p cohomology (i.e., the kernel is nilpotent and every element has some p-power power in the image) then H controls fusion in G; and therefore the inclusion is actually a cohomology isomorphism. Ellen Henke recently provided the final fusion theoretic argument that completes the purely algebraic proof of this stronger statement.
• 16:00, Room U2-205
  Jon Carlson (Athens, Georgia): Thick subcategories of the bounded derived category
  Abstract: This is joint work with Srikanth Iyengar. It is all about using methods from commutative algebra to study group representations. A new proof of the classification for tensor ideal thick subcategories of the bounded derived category, and the stable category, of modular representations of a finite group is obtained. The arguments apply more generally to yield a classification of thick subcategories of the bounded derived category of an artinian complete intersection ring. One of the salient features of this work is that it takes no recourse to infinite constructions, unlike the previous proofs of these results.

Friday, 10 February 2012

• 14:30, Room U2-205
  Jan Stovicek (Prague): Resolving subcategories for a commutative noetherian ring
  Abstract: Given a commutative notherian ring R, we classify resolving subcategories of mod-R consisting of modules of bounded projective dimension in terms of certain descending sequences of specialization closed subsets of the Zariski spectrum of R. This is a consequence of a similar classification result for cotilting classes in the category of infinitely generated modules over R. The talk is an account on joint work with L. Angeleri, D. Pospisil and J. Trlifaj.
• 16:00, Room U2-205
  Markus Perling (Bochum): Equivariant resolutions and maximal Cohen-Macaulay modules over affine toric varieties
  Abstract: We present a combinatorial framework which allows to translate free resolutions of Z^n-graded modules over the polynomial ring in n variables into projective resolutions over certain incidence algebras. We present two applications: 1. we explicitly determine the Betti-numbers and local cohomologies of certain such modules related to hyperplane arrangements; 2. we produce new examples of MCM modules over certain toric rings.

Friday, 27 January 2012

• 14:30, Room V2-213
  Hongmei Zhao (Nankai): On the structure of the augmentation quotient groups for some nonabelian groups
  Abstract: Let G be a finite group, ZG its integral group ring and delta^n(G) the n'th power of the augmentation ideal delta(G), denote Q_n(G)=delta^n(G)/delta^{n+1}(G) the augmentation quotient groups of G. We consider the dihedral group D_{2^tk}(k odd) and m'th symmetric group S_m, we show Q_n(D_{2^tk}) is an elementary 2-group and its rank is no more than 2t+1. As for Q_n(S_m), we have Q_n(S_m) is isomorphic with Z_2.
• 16:00, Room V2-213
  Barbara Baumeister (Bielefeld): Some aspects of group theory 200 years after Galois
  Abstract: Galois introduced the notion of a group to solve old geometric questions. In the talk I will show that this concept is still alive by discussing recent results on twin trees as well as on permutation polytopes.

Friday, 20 January 2012

• 13:15, Room V2-213
  Shoham Shamir (Bergen): A colocalization spectral sequence
  Abstract: Colocalization is a right adjoint to the inclusion of some subcategory. Given a differential graded algebra R, it is natural to ask for a spectral sequence which connects a colocalization in the derived category of R-modules and an appropriate colocalization in the derived category of graded modules over the cohomology ring of R. It turns out that, under suitable conditions, such a spectral sequence exists. This generalizes the Greenlees spectral sequence. I will describe this generalization and show some applications.
• 14:30, Room V2-213
  Irakli Patchkoria (Bonn): On the algebraic classification of module spectra
  Abstract: For any S-algebra R whose homotopy ring is sufficiently sparse and has graded global homological dimension less or equal than three, we construct an equivalence between the derived category of R and the derived category of its homotopy ring. This improves Bousfield-Wolbert algebraic classification of isomorphism classes of objects in the derived category of R. In the case of global dimension two, the p-local real connective K-theory, the first Johnson Wilson spectrum E(2) and the truncated Brown-Peterson spectrum BP<1>, for an odd prime p, serve as our main examples. Examples of S-algebras with three dimensional homotopy ring to which our result applies are E(3) and BP<2> at a prime greater or equal than five.

Friday, 16 December 2011

• 14:30, Room V2-213
  Claus Michael Ringel (Bielefeld): Representations of a quiver over the algebra of dual numbers
  Abstract: The representations of a quiver Q over a field k have been studied for a long time, and one knows quite well the structure of the category of kQ-modules. It seems to be worthwhile to consider also representations of Q over arbitrary finite-dimensional k-algebras A. The lecture will draw the attention to the case when A = k[epsilon] is the algebra of dual numbers, thus to the category of Lambda-modules, where Lambda = kQ[epsilon] is the path algebra of Q over A. The algebra Lambda is a 1-Gorenstein algebra, thus the torsionless Lambda-modules are known to be of special interest (as the Gorenstein-projective or maximal Cohen-Macaulay modules). They form a Frobenius category L, thus the corresponding stable category is a triangulated category T. This category T is triangle equivalent to the orbit category of the bounded derived category of the kQ-modules modulo the shift. The homology functor H yields a bijection between the indecomposables in T and those in mod kQ, the inverse is given by forming the minimal L-approximation. We also describe the embedding of the Auslander-Reiten quiver of mod kQ into that of T.
  
  This is a report on current joint investigations with Zhang Pu (SJTU).
• 16:00, Room V2-213
  Gena Puninski (Moskow): The Ziegler spectrum of domestic string algebras (Part II)
  Abstract: The Ziegler spectrum of a ring is a topological space whose points are indecomposable pure injective modules and basic open sets are determined by morphisms between finitely presented modules. This space captures a lot of information on the category of modules and there are many applications of this notion.
  In this series of talks we will describe some general method of classifying indecomposable p.i. modules and apply it to describe the Ziegler spectrum of string algebras. In the case of 1-domestic string algebra we will completely classify the points and the topology of Ziegler spectrum, confirming in particular a conjecture made by Ringel. We will also hope to prove that the Krull-Gabriel dimension of any 1-domestic string algebra does not exceed 3 (Schroer's conjecture).
  We will discuss a general approach to classifying arbitrary (not necessarily indecomposable) pure injective modules over string algebras, most prominent of those are superdecomposable modules, and formulate some global conjectures on their structure.
  The lectures will be given in `proof by example' style, with many diagrams drawn; no previous knowledge of the Ziegler spectrum is expected, though some acquaintance with representation theory will help.

Friday, 02 December 2011

• 13:15, Room V2-213
  Artem Lopatin (Omsk): Matrix identities with forms
  Abstract: A linear group GL(n) acts on d-tuples of n x n matrices by simultaneous conjugation. The algebra R(n,d) of polynomial invariants of this action is called the algebra of matrix GL(n)-invariants. In the case of arbitrary characteristic of the base field Donkin [Invent. Math. 110 (1992), 389-401] established generators of R(n,d) and Zubkov [Algebra and Logic 35 (1996), No. 4, 241-254] described relations between them. Namely, Zubkov showed that the ideal of relations is generated by the coefficients $\sigma_k$ of the characteristic polynomial of a matrix for k>n. We proved that the ideal of relations is actually generated by $\sigma_k$ for $n<k\leq 2n$. In particular, we showed that the T-ideal of identities of $M_n$ with forms is finitely based.
• 14:30, Room V2-213
  Grzegorz Bobinski (Torun): Semi-invariants of concealed-canonical algebras
  Abstract: The descriptions of the algebras of semi-invariants for regular dimension vectors over Euclidean and canonical algebras are basically identical. Thus it is natural to expect that these descriptions generalize to arbitrary concealed-canonical algebras. The main result of my talk states that this is really the case.
• 16:00, Room V2-213
  Gena Puninski (Moskow): The Ziegler spectrum of domestic string algebras (Part I)
  Abstract: The Ziegler spectrum of a ring is a topological space whose points are indecomposable pure injective modules and basic open sets are determined by morphisms between finitely presented modules. This space captures a lot of information on the category of modules and there are many applications of this notion.
  In this series of talks we will describe some general method of classifying indecomposable p.i. modules and apply it to describe the Ziegler spectrum of string algebras. In the case of 1-domestic string algebra we will completely classify the points and the topology of Ziegler spectrum, confirming in particular a conjecture made by Ringel. We will also hope to prove that the Krull-Gabriel dimension of any 1-domestic string algebra does not exceed 3 (Schroer's conjecture).
  We will discuss a general approach to classifying arbitrary (not necessarily indecomposable) pure injective modules over string algebras, most prominent of those are superdecomposable modules, and formulate some global conjectures on their structure.
  The lectures will be given in `proof by example' style, with many diagrams drawn; no previous knowledge of the Ziegler spectrum is expected, though some acquaintance with representation theory will help.

Friday, 25 November 2011

• 16:00, Room C0-269
  Christopher Voll (Bielefeld): Representation zeta functions of arithmetic groups
  Abstract: A group is called (representation) rigid if it has, for each n, only finitely many irreducible complex representations of dimension n. To study the representation growth of rigid groups is to study the arithmetic and asymptotic properties of the number of such representations, as n tends to infinity. If these numbers grow at most polynomially, a profitable approach to their study is to encode them in a Dirichlet generating series - the group's representation zeta function. Under additional assumptions, such zeta functions have Euler products indexed by places in algebraic number fields. The factors of such Euler products can be studied using a wealth of methods from geometry and combinatorics. Major questions regarding representation zeta functions of groups ask about properties of the Euler factors, such as rationality, and local and global abscissae of convergence.
  
  I will report on recent joint work on representation zeta functions of arithmetic groups. In a project with Avni, Klopsch and Onn, we establish a conjecture of Larsen and Lubotzky on the abscissae of convergence of irreducible lattices in higher-rank semisimple groups. In joint work with Stasinski we give uniform formulae for representation zeta functions of finitely generated nilpotent groups. A common feature of both projects is the use of sophisticated machinery from the theory of p-adic integration and a Kirillov orbit method to parameterize representations by co-adjoint orbits.

Friday, 18 November 2011

• 13:15, Room V2-213
  Sarah Scherotzke (Bonn): The Integral Cluster Category
  Abstract: In my talk, we will consider the question when orbit categories of triangulated categories are again triangulated. I will present some examples where this fails and give a sufficient condition proven by Bernhard Keller for the orbit category of a triangulated category to have a natural triangulated structure. Applying this result to the Cluster category associated to a finite acyclic quiver over a field shows that it is triangulated. In joint work with Bernhard Keller, we proved that the Cluster category defined over certain commutative rings are triangulated, we classify the Cluster-tilting objects and show that they are linked by mutation. The proof in the integral case does not use Keller's criteria and requires a different approach of which I will give sketch.
• 14:30, Room V2-213
  Roger Wiegand (Lincoln): Brauer-Thrall theorems and conjectures for commutative local rings
  Abstract: The Brauer-Thrall Conjectures, now theorems, were originally formulated in terms of representations of finite-dimensional algebras. They say, roughly speaking, that failure of finite representation type entails the existence of lots of indecomposable representations of large dimension. These conjectures have been successfully transplanted to the representation theory of commutative local rings. This talk will be a survey of such results, conjectures and counterexamples, for various categories of finitely generated modules over a commutative Noetherian local ring. The emphasis will be on maximal Cohen-Macaulay modules over Cohen-Macaulay local rings.
• 16:00, Room V2-213
  Sylvia Wiegand (Lincoln): The Anatomy of a Stranger: A (Barely) non-Noetherian Ring
  Abstract: In ongoing work with William Heinzer and Christel Rotthaus over the past twenty years we have been applying a construction technique for obtaining sometimes baffling, sometimes badly behaved, sometimes Noetherian, sometimes non-Noetherian integral domains. This technique of intersecting fields with power series rings goes back to Akizuki in the 1930s and Nagata in the 1950s; since then it has been employed by Nishimuri, Heitmann, Ogoma, the authors and others.
  In particular we show how to obtain a three-dimensional near-Noetherian unique factorization domain B that is tantalizingly close to being Noetherian but is not quite, because exactly one prime ideal has height two and it is the only nonfinitely generated prime ideal of B. The unique maximal ideal of B is 2-generated. We also mention more mysterious generalizations to higher dimensions.

Friday, 04 November 2011

• 14:30, Room V2-213
  Yong Jiang (Bielefeld): Parametrizations of canonical bases and irreducible components of nilpotent varieties
  Abstract: It is known that the set of irreducible components of nilpotent varieties provides a geometric realization of the crystal basis. For each reduced expression of an element in the Weyl group, Geiss, Leclerc and Schroeer have recently given a parametrization of the set of irreducible components in studying the cluster structure of the coordinate ring of the corresponding unipotent subgroup. We show that their parametrization is compatible with Lusztig's parametrization of canonical basis. And we also give some interpretations of Lusztig's transition maps.
  
• 16:00, Room V2-213
  Moritz Groth (Nijmegen): On the theory of derivators
  Abstract: The theory of derivators -- going back t Grothendieck and Heller -- is a purely (2-)categorical approach to an axiomatic homotopy theory. The usual passage from a model category (resp. an abelian category) to the underlying homotopy category (resp. derived category) result in a loss of information. The typical defects of triangulated categories (e.g. the non-functoriality of the cone construction) can be seen as a reminiscent of this fact. The basic idea of a derivator is that one should instead simultaneously form homotopy/derived categories of `all' diagra categories and also keep track of the restriction and homotopy Kan extensio functors. The aim of this talk is to give an introduction to the theory of derivators and t (hopefully) advertise it as a convenient 'weakl terminal' approach to axiomatic homotopy theory.
  

Friday, 28 October 2011

• 14:30, Room V2-213
  Anna-Louise Paasch (Bielefeld): Monoid algebras of projection functors
  Abstract: For each simple representation of a finite acyclic quiver there is a projection functor onto the kernel of the covariant Hom-functor of the simple. We study the monoid (algebra) generated by those projection functors w.r.t. composition.
  The monoid algebra associated with a linearly oriented Dynkin quiver of type A is discussed in detail. We determine defining relations for several other cases and illustrate the influence of the orientation rather than the representation type of the underlying graph.
• 16:00, Room V2-213
  Ralf Meyer (Göttingen): Hereditary exact categories from equivariant bivariant K-theory
  Abstract: Together with Ryszard Nest and my students Rasmus Bentmann and Manuel Köhler, I have been studying Universal Coefficient Theorems in bivariant K-theory for C*-algebras. The idea there is to find a K-theoretic invariant that completely classifies certain diagrams of C*-algebras up to weak equivalence. With a guess for this invariant in hand, a general machinery for homological algebra in triangulated categories provides the required Universal Coefficient Theorem provided certain modules over a certain ring have projective resolutions of length 1. As a result, we proved several positive and negative results about the existence of such resolutions for certain rings.
  The results we obtained hint at a connection with quiver representations, in particular, the special features of ADE-quivers, but we do not yet understand this. I hope that discussions with algebraists can clarify this relationship and what kind of results to expect in more general cases.

Friday, 21 October 2011

• 13:15, Room V2-213
  Jesse Burke (Bielefeld): Finite injective dimension over rings with Noetherian cohomology
  Abstract: After discussing rings with Noetherian cohomology, examples of which include group rings of finite groups and complete intersection rings, we will state and prove a criterion for a complex over such a ring to have finite injective dimension. The criterion generalizes a theorem in the representation theory of finite groups and a theorem of Avramov-Buchweitz for complete intersection rings. The proof uses the support theory of Benson, Iyengar, and Krause.
  
• 14:30, Room V2-213
  Philipp Lampe (Bielefeld): Cluster theory and Lusztig's canonical basis
  Abstract: With a Dynkin quiver one associates two triangulated categories: the cluster category and stable category of the module category of the preprojective algebra. We will explain why the categories provide an additive categorification of cluster algebras. They are related to various bases of the universal enveloping algebra of the same type. The talk concerns especially the connection between cluster variables and the dual canonical basis.
• 16:00, Room V2-213
  Robert Marsh (Leeds): Cluster presentations of reflection groups (joint work with Michael Barot, UNAM, Mexico)
  Abstract: We give a presentation of a finite crystallographic reflection group in terms of an arbitrary seed in the corresponding cluster algebra of finite type.

Friday, 07 October 2011

• 14:15, Room V4-119
  Gil Kaplan (Tel Aviv): Some new results on finite T-groups
  Abstract: A group is called a T-group if all its subnormal subgroups are normal, or, equivalently, if normality is a transitive relation among its subgroups. The subject was extensively studied since the seminal paper by Gaschuetz (1957), in which he described the structure of finite solvable T-groups. We give new results in this subject and new characterizations of solvable T-groups. One characterization involves properties of maximal subgroups. A second characterization involves product of conjugate subgroups.

For information on earlier talks please check the complete seminar archive.

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• Last file change on 22 Dec 2011 by Andre Beineke
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